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I am looking for a holomorphic action of $C_{2}\times C_{2}$ on $\mathbb{P}^1$. Is it true that there is a unique effective action given by $$ a:z\rightarrow -z, \ \ \ \ b:z\mapsto 1/z $$ up to change of variable? Here $a$ is the generators of the first (second) factor of $C_{2}\times C_{2}$.

What about $C_{3}\times C_{3}$? I could not find any effective action.

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Isn't $z=-z$ in $\Bbb P^n$? In which case $a$'s action is trivial. Or am I missing somehting? – Arthur Sep 18 '12 at 9:00
He uses $z$ as an inhomogeneous coordinate $[z,1]\in \mathbb{P}^1$. – M. K. Sep 18 '12 at 9:29

It is true. Any finite group of automorphisms can be conjugated to that it becomes a subgroup of the group of rotations of the Riemann sphere, and there is a classification of finite subgroups (up to conjugation) of $SO(3)$, containing $C_2\times C_2$ just once (as the group of rotations by $\pi$ around the coordinate axes).

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